On the existence of a first order phase transition at small vacuum angle $\theta$ in the $C{P}^3$ model

We examine the phase structure of the $C{P}^3$ model as a function of $\theta$ in the weak coupling regime. It is shown that the model has a first order phase transition at small $\theta$. We pay special attention to the extrapolation of the data to the infinite volume. It is found that the critical value of $\theta$ decreases towards zero as $\beta$ is taken to infinity.